Question

Let A be a square matrix defined by

                               

(a) Find the eigenvalues and eigenspaces of A.

(b) Show that A is not diagonalizable but triangularizable. Triangularize A.

Answer 1

Solution

(a) Find the eigenvalues and eigenspaces of A.

Find the eigenspaces associated to eigenvalues

Therefore, the eigspace associated with 

(b) Show that A is not diagonalizable but triangularizable. Triangularize A.

Therefore, A is not diagnoalizable, but triangularizable

(c) Solve the system of linear differential equation dx/dt=Ax

The solution of system x'(t)=A x(t) is

Then the solution to x(t) is

Then, solve the initial value problem x'(t)=Ax(t)+B(t)

Determine the constants of particular solution first where\